How can I make an existing mesh appear to be drawn in animation?
You can use animate the size of another object (e.g. the radius of a cylinder) using keyframes, and then apply a boolean intersect modifier to your mesh.



Step by step


Create a cylinder with the middle in the point where you want the "drawing" to start. Make sure that it's thick enough to cover all the mesh on the Z direction



Scale it down on the X and Y directions hitting S then Shift+Z


Set a keyframe to the current size using I > Scaling. Then move to the end frame, scale it up (again only along X-Y with S and Shift+Z), and set another keyframe (I > Scaling).


Optionally, you can avoid the visualization of a solid cylinder by setting its maximum draw type to wire


Last, select your mesh and add a Boolean modifier, using the Intersect method and your cylinder as the reference object